Simplify. Express Your Answer Using Positive Exponents.${ 7pq \cdot 4p^6q \cdot 8pq^2 }$

Mar 9, 2025 by ADMIN 90 views

Simplify Expressions Using Positive Exponents

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Understanding Exponents and Multiplication

When dealing with expressions that involve exponents and multiplication, it's essential to understand the rules of exponentiation and how they apply to different variables. In this article, we'll explore how to simplify expressions using positive exponents, focusing on the given expression: $7pq \cdot 4p^6q \cdot 8pq^2$ .

What are Exponents?

Exponents are a shorthand way of representing repeated multiplication of a number or variable. For example, $a^2$ means $a \cdot a$ , and $a^3$ means $a \cdot a \cdot a$ . When we multiply variables with exponents, we add the exponents. This rule is known as the product of powers rule.

The Product of Powers Rule

The product of powers rule states that when we multiply two variables with exponents, we add the exponents. Mathematically, this can be represented as:

$a^m \cdot a^n = a^{m+n}$

This rule applies to both numbers and variables.

Simplifying the Given Expression

Now that we understand the product of powers rule, let's apply it to the given expression: $7pq \cdot 4p^6q \cdot 8pq^2$ . To simplify this expression, we'll first multiply the coefficients (the numbers in front of the variables) and then apply the product of powers rule to the variables.

Multiplying the Coefficients

The coefficients are 7, 4, and 8. To multiply them, we simply multiply the numbers together:

$7 \cdot 4 \cdot 8 = 224$

Applying the Product of Powers Rule

Now, let's focus on the variables. We have $pq$ , $p^6q$ , and $pq^2$ . To simplify these variables, we'll apply the product of powers rule:

$pq \cdot p^6q \cdot pq^2 = p^{1+6+1}q^{1+1+2}$

Using the product of powers rule, we add the exponents:

$p^{1+6+1} = p^8$ $q^{1+1+2} = q^4$

Combining the Simplified Variables and Coefficients

Now that we've simplified the variables and coefficients, we can combine them to get the final simplified expression:

$224p^8q^4$

Conclusion

In this article, we've explored how to simplify expressions using positive exponents. We started by understanding the product of powers rule, which states that when we multiply two variables with exponents, we add the exponents. We then applied this rule to the given expression: $7pq \cdot 4p^6q \cdot 8pq^2$ . By multiplying the coefficients and applying the product of powers rule to the variables, we simplified the expression to $224p^8q^4$ .

Real-World Applications of Exponents

Exponents have numerous real-world applications, including:

Finance: Exponents are used to calculate compound interest and investment returns.
Science: Exponents are used to describe the growth and decay of populations, chemical reactions, and physical phenomena.
Engineering: Exponents are used to calculate stress and strain in materials, as well as to model population growth and decay.

Common Mistakes to Avoid

When simplifying expressions using positive exponents, it's essential to avoid common mistakes, including:

Forgetting to multiply the coefficients: Make sure to multiply the coefficients together before applying the product of powers rule.
Not adding the exponents correctly: Double-check that you're adding the exponents correctly when applying the product of powers rule.
Not simplifying the expression fully: Make sure to simplify the expression fully, including combining like terms and eliminating any unnecessary variables.

Tips for Simplifying Expressions

To simplify expressions using positive exponents, follow these tips:

Start by multiplying the coefficients: Multiply the coefficients together before applying the product of powers rule.
Apply the product of powers rule carefully: Make sure to add the exponents correctly when applying the product of powers rule.
Simplify the expression fully: Make sure to simplify the expression fully, including combining like terms and eliminating any unnecessary variables.

Conclusion

In conclusion, simplifying expressions using positive exponents is a crucial skill in mathematics. By understanding the product of powers rule and applying it correctly, we can simplify complex expressions and arrive at the final answer. Remember to multiply the coefficients together, apply the product of powers rule carefully, and simplify the expression fully to avoid common mistakes. With practice and patience, you'll become proficient in simplifying expressions using positive exponents.

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Frequently Asked Questions

In this article, we'll address some of the most frequently asked questions about simplifying expressions using positive exponents.

Q: What is the product of powers rule?

A: The product of powers rule states that when we multiply two variables with exponents, we add the exponents. Mathematically, this can be represented as:

$a^m \cdot a^n = a^{m+n}$

Q: How do I apply the product of powers rule?

A: To apply the product of powers rule, simply add the exponents of the variables. For example, if we have $a^2 \cdot a^3$ , we would add the exponents to get $a^{2+3} = a^5$ .

Q: What if I have multiple variables with exponents?

A: If you have multiple variables with exponents, you can apply the product of powers rule to each variable separately. For example, if we have $a^2 \cdot b^3 \cdot c^4$ , we would add the exponents of each variable to get $a^2 \cdot b^3 \cdot c^4 = a^2 \cdot b^3 \cdot c^4$ .

Q: Can I simplify expressions with negative exponents?

A: Yes, you can simplify expressions with negative exponents. To do this, you can use the rule that $a^{-n} = \frac{1}{a^n}$ . For example, if we have $a^{-2}$ , we can rewrite it as $\frac{1}{a^2}$ .

Q: How do I simplify expressions with variables and numbers?

A: To simplify expressions with variables and numbers, you can apply the product of powers rule to the variables and then multiply the numbers together. For example, if we have $2a^2 \cdot 3b^3$ , we would add the exponents of the variables to get $2a^2 \cdot 3b^3 = 6a^2b^3$ .

Q: What if I have a fraction with exponents?

A: If you have a fraction with exponents, you can simplify it by applying the product of powers rule to the numerator and denominator separately. For example, if we have $\frac{a^2}{b^3}$ , we would add the exponents of the numerator and denominator to get $\frac{a^2}{b^3} = \frac{a^2}{b^3}$ .

Q: Can I simplify expressions with radicals?

A: Yes, you can simplify expressions with radicals. To do this, you can use the rule that $\sqrt[n]{a^n} = a$ . For example, if we have $\sqrt[3]{a^3}$ , we can rewrite it as $a$ .

Common Mistakes to Avoid

When simplifying expressions using positive exponents, it's essential to avoid common mistakes, including:

Forgetting to multiply the coefficients: Make sure to multiply the coefficients together before applying the product of powers rule.
Not adding the exponents correctly: Double-check that you're adding the exponents correctly when applying the product of powers rule.
Not simplifying the expression fully: Make sure to simplify the expression fully, including combining like terms and eliminating any unnecessary variables.

Tips for Simplifying Expressions

To simplify expressions using positive exponents, follow these tips:

Start by multiplying the coefficients: Multiply the coefficients together before applying the product of powers rule.
Apply the product of powers rule carefully: Make sure to add the exponents correctly when applying the product of powers rule.
Simplify the expression fully: Make sure to simplify the expression fully, including combining like terms and eliminating any unnecessary variables.

Conclusion

Additional Resources

For more information on simplifying expressions using positive exponents, check out the following resources:

Math textbooks: Consult a math textbook for a comprehensive explanation of the product of powers rule and how to apply it.
Online resources: Visit online resources such as Khan Academy, Mathway, or Wolfram Alpha for interactive lessons and practice problems.
Practice problems: Practice simplifying expressions using positive exponents with online practice problems or worksheets.

By following these tips and resources, you'll become proficient in simplifying expressions using positive exponents and be able to tackle complex math problems with confidence.